If hx = x^7 and gx = x^2, which expression is equivalent to gh5?
To find which expression is equivalent to gh5, we start by substituting the given functions into the expression. We know that: hx = x^7 gx = x^2 Now, to find gh5, we need to evaluate g(5) and h(g(5)). First, let’s calculate g(5): g(5) = 5^2 = 25. Next, we need to find h(g(5)), which is […]
Find the Area of the Region in the First Quadrant Enclosed by the X-Axis, the Line y = x, and the Circle x² + y² = 32
To find the area of the region in the first quadrant enclosed by the x-axis, the line y = x, and the circle x² + y² = 32, we will follow these steps: 1. **Identify the Intersections:** We first need to determine where the line intersects the circle. We substitute y = x into the […]
Which algebraic expression is a trinomial?
A trinomial is an algebraic expression that consists of three terms. To determine if the given expression qualifies as a trinomial, we first need to break down the expression provided: Let’s simplify the expression: x³ + x² + 1 + 2x³ + x² + 4x³ + x² + 1x + 6x + 6 Now, combine […]
What is an equation for the translation x 22 y 12 16 by 4 units left and 6 units up?
To find the new equation after translating the point (22, 12, 16) by 4 units to the left and 6 units up, we start with the original coordinates of the point. The translation can be described as follows: Translating 4 units to the left involves subtracting 4 from the x-coordinate. Translating 6 units up involves […]
Verify the Identity: sin(x) * sin(y) = sin(x) * y + 2 * cos(x) * sin(y)
To verify the identity, we need to analyze and simplify both sides of the equation. Let’s start with the left-hand side (LHS): LHS = sin(x) * sin(y) Now, let’s look at the right-hand side (RHS): RHS = sin(x) * y + 2 * cos(x) * sin(y) We need to manipulate the left-hand side to see […]
When to Use Law of Sines vs Law of Cosines
The Law of Sines and the Law of Cosines are both essential tools in trigonometry for solving triangles, but they are used in different situations. Law of Sines is best used when you have: Two angles and one side (AAS or ASA). Two sides and a non-included angle (SSA). For example, if you know two […]
What is the average speed of a train traveling at different speeds?
To find the average speed of the train, we need to calculate the total distance traveled and the total time taken during the entire journey. 1. **First Segment:** The train travels at a speed of 60 km/h for 0.52 hours. Distance = Speed × Time = 60 km/h × 0.52 h = 31.2 km 2. […]
What happens to a ball dropped from a height of 10 feet that bounces to 34% of its previous height after each bounce?
When a ball is dropped from a height of 10 feet, it falls to the ground due to gravity and then bounces back up. In this scenario, each time the ball hits the floor, it bounces back to 34% of the height from which it fell. For the first bounce, the height will be: Height […]
Identify the Five Number Summary of the Following Data Set: 7, 11, 21, 28, 32, 33, 37, 43
The five number summary of a data set consists of five important descriptive statistics: the minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum. Let’s break down the calculation for the provided data set. Given data set: 7, 11, 21, 28, 32, 33, 37, 43 Minimum: This is the smallest number in the […]
Consider the following function f(x) = 1 / x^23; find f(1) and f'(1)
To find f(1), we simply substitute x = 1 into the function: f(1) = 1 / (1)^23 = 1 Now, to find f'(x), we need to differentiate the function f(x). The function can be rewritten as: f(x) = x-23 Using the power rule of differentiation, which states that d/dx[x^n] = n*x^(n-1), we can differentiate: f'(x) […]